Application of Gamma functions to the determination of 2 unit hydrographs 3

8 * Correspondence: lihongyan@jlu.edu.cn 9 10 Abstract:There are many methods for calculating unit hydrograph, such as analysis method, trial algorithmand 11 least squares method. But these methods have certain requirements for flood datas and the unit hydrograph may 12 not be optimal. Based on the theory of composition, a hydrological system was viewed as a generalizedcollection 13 in this study and Gamma functions were used to simulate the basin convergence process. At the same time,the 14 Gamma function is parameterized and the parameters of Gamma function are optimized by geneticalgorithm, 15 which is based on the minimum error between the calculation of confluence process and the measurement process, 16 before deriving the unit hydrograph. The Collins iteration method was used to compute the unit hydrograph. The 17 results of actual calculated examples showe that this method is more precise than other methods, while it canalso 18 illustrate the law of runoff. 19


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The unit hydrograph is an important method for simulating the flow concentration of a 23 conceptual hydrological model; it was proposed by Sherman (1932). In actual applications, the 24 derivation of the unit hydrograph is still an important component when forecasting the basin 25 rainfall and runoff.

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A unit hydrograph ( Viessman, 1989;Raghunath, 2006) refers to the unit net constant 27 rainfall uniformly distributed over a watershed of unit surface and for unit duration. Periods of 1, 3, 28 6, and 12 h can be selected and the unit rainfall (runoff depth) is generally 10 mm. The actual net 29 rainfall often does not equal 1 unit for these time periods, so it is necessary to make two basic hydrograph confluence takes the basin as a whole and assumes that the net rain is uniformly 46 distributed over the whole basin, without considering the inhomogeneity within the system; the 47 basin confluence system is a linear time-invariant system, and at the same time, it is viewed that 48 the net rainfall and the formation of the flow process are in agreement to superposition 49 relationship. Therefore, the essential characteristics of the unit hydrograph are lumped resistance, 50 linearity, and time invariance.

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Conceptually, the unit hydrograph is a linear time-invariant basin system with a convergent 52 flow curve. However, the physical mechanism of the watershed conflux is not considered by the 53 method used to derive the unit hydrograph (Ramirez, 2000). The principle used to calculate the 54 unit hydrograph is based on the system input (rainfall), which is converted using the unit 55 hydrograph to determine the system response output (outlet section flow process), where the error 56 is minimized. The traditional methods of derivation are as follows.

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Analytical approach: the basin outlet section of the surface runoff is Q 1 , Q 2 , , Q l , the 58 rainfall process is h 1 , h 2 , , h m , where Eqn. (1) comprises q 1 , q 2 , ,q n unknown linear 59 algebraic equations. The solutions of the equations can be obtained using a unit hydrograph.
where n is the number of periods of the unit hydrograph and n  l  m 1 .

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In theory, there are no errors with the analytical approach when using the rainfall runoff 63 measurements. This approach can obtain the correct answer if the watershed conflux conforms to a 42 https://doi.org/10.5194/npg-2020-1 Preprint.

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There are many methods for determining unit hydrographs, e.g., the Z transform method and 83 the harmonic analysis method (Dooge, 1973  studying the composition of things. This theory considers the analysis of three concepts, i.e., the 116 general set, the distribution function, and the degree of complexity. This theory is also considered 117 the most highly approved principle followed by random systems, i.e., the entropy principle.

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The variable x is continuous and random, and can be viewed as a general set of flag 119 variables. If the pdf f (x) of x agrees with the following function: Then the pdf follows a Gamma distribution, where  and k are shape and scale parameters.
122 This is one of the famous Pearson pdfs, which is known as a Pearson type III distribution. The 123 curve has a peak with a left-right asymmetry. In nature, many phenomena follow this distribution.

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In China, hydrological studies often use the Pearson type III distribution to simulate hydrological 125 data series, because it has a greater than or equal to zero lower bound on the variable requirements 126 and its elasticity is greater than the normal distribution (Ye and Xia, 2002). This choice is based 127 on experience, but it lacks a theoretical justification.

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In this study, f (x) is the pdf of a positively defined random variable, i.e., while v is the geometric mean of the random variable x , which can be expressed as the 141 algebraic average of the logarithm, i.e., The entropy of the random variable x can be written as

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Given the constraints in Eqns. (4), (5), and (6), the Lagrange method can be used to estimate 146 the distribution function F based on the maximum entropy to determine the distribution 147 function. Thus, F is defined as follows: Where, C 1 , C 2 , and C 3 are undetermined constants. The entropy principle demands that the 150 151 value of F is maximal. The partial derivative of obtained using Eqn (8). The results are as follows.
f () , i.e., the partial derivative is 0, can be 153 This formula can be used to obtain the distribution function. It is the product of the power 154 function and the exponential function, and its form is identical to a Gamma function.

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Hydrological data are random variables that exceed zero. If the hydrological processes are  we should determine an initial value, an iteration function and a restriction condition according to 203 the actual situation and data firstly, until the absolute value of the initial value and the calculated 204 approximation value is less than a certain value.That is to say, we find the exact desired value.

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2 Approach used to determined the unit hydrograph 206 The overall calculation process is divided into two parts: (1) the parameters of the unit 207 hydrograph are optimized using the genetic algorithm, so the initial unit hydrograph can be 208 calculated; and (2) the final unit hydrograph is calculated using the Collins iterative method.  The variables x 1 , point value.
x 2 , and 219 If one chromosome is v 220 the unit hydrograph), the unit hydrograph obtained using the constant given above is as Using Eqn. x * x * x *

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A set of parameters, 1 , 2 , and 3 , are searched based on the following conditions. The is calculated by the optimum parameters, as follows: is deduced, and new i is deduced continuously by In Table 1, the data were taken from a previous study (Zhuang and Lin, 1986). The unit hydrographs determined using the two methods are shown in columns (5) and (7) in 262 Table 1. A comparison of the unit hydrographs is shown in Figure 4. The flow processes 263 calculated using the unit hydrographs are shown in columns (4) and (6) in Table 1. A comparison   264 between the calculated flow process and the measured flow is shown in Figure 5. The actual hydrological data in Table 1 show that the period number for the runoff was 5 and  Table 2. 277 Table 2 The error statistics of example 1

Project Method GACIM The trial and error
The error of flood peak(m 3 /s) 0 10 The maximum error of discharge(m 3 /s) 212 216 The average absolute error of discharge(m 3 /s) 37.31 46.19 The total error of flood peak discharge(m 3 /s.h) -111 -51 The relative error of flood peak discharge (%) -1.20 -0.55 278 279 Example 2

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In Table 3, the data were taken from a previous study (Li and Zheng, 1982). (m 3 /s) The unit hydrographs determined using the two methods are shown in columns (5) and (7) in 284 Table 3. A comparison of the unit hydrograph is shown in Figure 6. The flow processes calculated 285 using the unit hydrographs are shown in columns (4) and (6) in Table 3 Figure 4 shows that GACIM was significantly better than the trial and error method in terms 292 of the shape of the curve. We also compared the flow process in the outlet section and the data 293 obtained using the two unit hydrographs. A statistical analysis of the results is shown in Table 4.
294 Table 4 The error statistics of example 2

Project Method GACIM The trial and error
The error of flood peak(m3/s) 0 -3 The maximum error of discharge(m3/s) 11 -12 The average absolute error of discharge(m3/s) 1.69 -0.23 The total error of flood peak discharge(m3/s.h) 22 -3 The relative error of flood peak discharge (%) 1.25 -0.17 295 296 From Table 2 and Table 4, the calculation accuracy of BGACM is obviously better than 297 that of trial-and-error method in most projects. Although the total error of flood volume is larger 298 than that of trial-and-error method, the relative error of flood volume is only 1.2% and 1.25%, so it 299 does not affect the application of actual projects.
300 Figure 6 and 7 show that the flow processes of the two unit hydrographs were similar.

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However, a comparison of the shapes of the unit hydrograph showed that the continuity and 302 smoothness of GACIM were better than the trial and error method. The GACIM method 303 conformed better with the features of a time-invariant system.

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It can be seen that BGACM method is better at simulating river basin confluence process, 305 which depends on the physical mechanism of the algorithm, while trial-and-error method pays 306 more attention to the balance of total flood volume. This is the respective characteristics and 307 advantages of the two algorithms exactly.

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Other methods for calculating unit hydrographs include the analysis method, least squares 320 method, and the trial and error method. These methods are more focused on the unit hydrograph as 321 an outlet flow process and they fit the measured flow precisely, but they ignore the composition 322 and structure of the unit hydrograph itself. Example 2 shows that GACIM performed better at 323 simulating the basin confluence process, whereas other methods paid more attention to the balance 324 of the total flood volume.

(b) Genetic operator design issues 326
A genetic algorithm is a very useful optimization tool. Its biggest advantage is that it has 327 wide adaptability and unlimited problem space, so it can handle many different constraints. This 328 strategy uses a penalty factor. This is because the genetic algorithm method delivers exhaustive 329 engineering accuracy if the population is sufficiently large.

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There are two types of genetic algorithm, i.e., the standard genetic algorithm (crossover and 331 mutation) and evolutionary computing (selection). A genetic algorithm simulates the 332 recombination of genes to create new offspring in each generation, whereas evolutionary 333 computation is a population process that updates each generation.

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In this study, a genetic algorithm was used to optimize the parameters of the Gamma function 335 and the unit hydrograph was calculated according to the law of basin confluence. Thus, the 336 parameters were generated by a genetic algorithm. Therefore, the design of the genetic operators is 337 related directly to whether reasonable generation parameters could be obtained.

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A genetic algorithm has two components: crossover and mutation. Crossover is the main 339 genetic operation that generates new individuals, but it also maintains the relative stability of the 340 population at the same time. However, the variation is a basic calculation and the main effect is to 341 produce a new gene from the population, which provides new information for the population.

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In general, the initial population of the genetic algorithm is generated in the value space.  In the first generation, the filial generation caused by the crossover operator was still in the 372 initial parameter space, which corresponds to the inner loop in Figure 8. However, the filial 373 generation caused by the mutation operator was beyond this range and it expanded to the second https://doi.org/10.5194/npg-2020-1 Preprint.

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Based on the theory of composition, the distribution function in statistical physics has been 393 extended to hydrology as a non-physics field. Thus, hydrological systems can be viewed as a 394 generalized collection. The regularities of hydrological phenomena have been simulated using 395 distribution functions. Distribution functions and functional relationships have been determined 396 using observation data, which generally means that objective laws are formalized.

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The present study was a preliminary attempt to investigate the quantitative relationships 398 among hydrological phenomena based on the theory of composition and its distribution function.

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The author believes that this theory could be a new approach to exploring hydrological rules.